The crucial role of proof--a classical defense against mathematical empiricism

The Crucial Role of Proof: A Classical Defense

Against Mathematical Empiricism

by

Catherine Allen Womack

Submitted to the Department of Linguistics and Philosophy

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

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March 10, 1993

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The Crucial Role of Proof: A Classical Defense Against

Mathematical Empiricism

by

Catherine Allen Womack

Submitted to the Department of Linguistics and Philosophy on March 10, 1993, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

Mathematical knowledge seems to enjoy special status not accorded to scientific knowledge: it is considered a priori and necessary. We attribute this status to math-ematics largely' because of the way we come to know it-through following proofs. Mathematics has come under attack from sceptics who reject the idea that mathe-matical knowledge is a priori. Many sceptics consider it to be a posteriori knowledge, subject to possible empirical refutation. In a series of three papers I defend the a priori status of mathematical knowledge by showing that rigorous methods of proof are sufficient to convey a priori knowledge of the theorem proved.

My first paper addresses Philip Kitcher's argument in his book The Natuire of

Mathematical Knowledge that mathematics is empirical. Kitcher develops a view of

a priori knowledge according to which mathematics is not a priori. I show that his

requirements for knowledge in general as well as a priori knowledge in particular

are far too strong. On Kitcher's view, some correct proofs may not even convey

knowledge, much less a priori knowledge. This consequence suggests that Kitcher's

conception of the a priori does not respond to properties of mathematics that have been responsible for the view that it is non-empirical.

In my second paper I examine Imre Lakatos' fallibilism in the philosophy of math-ematics. Lakatos argued that some mathematical propositions are subject to what he calls "refutations", by which he means to include falsification on extra-logical grounds. Lakatos cites Kalmar's scepticism about Church's Thesis as a case in point. I examine this case in detail, concluding that the failure of Lakatos' thesis in this prima facie favorable case casts doubt upon the thesis generally.

My third paper is a defense of the classical conception of proof against Thomas Tymoczko's thesis that only arguments that are surveyable by us can count as proofs. Tymoczko concluded from his thesis that the computer-assisted proof of the Four

Color Theorem involves an extension of the concept of proof hitherto available in mathematics. The classical conception regards the computer-assisted proof as a real proof, which we are unable to survey. Tymoczko recognizes that formalizability is a criterion for whether an argument is a proof, but he does not, in published work, note that formalizability and surveyability are often conflicting ideals. The classical theory recognizes both ideals because it regards the question whether something is a proof as distinct from the question of whether we can recognize it as such, or how confident we can be that it is one.

Thesis Supervisor: James Higginbotham Title: Professor

Acknowledgments

Completing this dissertation required only slightly less personnel, strategic planning and financial and emotional resources than the Allied landing at Normandy. Given this, it is unsurprising that I have many people to thank.

First of all, I want to thank my advisors Jim Higginbotham and George Boolos for providing a challenging and rigorous intellectual environment in which I could witness active philosophical work and engage in asking the tough questions of analytic philosophy. George's obvious joy in his work made me enthusiastic about philosophy of mathematics right from the start of my tenure at MIT; Jim's sense of humor helped make the monumental task of writing a thesis seem just a bit lighter.

Hilary Putnam encouraged me and gave me positive feedback on my ideas when I was feeling most discouraged. I feel priviledged to know him and have benefitted greatly from his brilliance and his kindness. Our weekly chats while I was his TA at Harvard helped motivate me to think seriously and work hard; I felt rejuvenated and excited about philosophy after talking with him.

Sylvain Bromberger has always been supportive, funny, sympathetic, stern when necessary, and right much of the time. I have been able to count on him when I needed an honest opinion.

I would be remiss not to thank the people who were responsible for helping get me into this business in the first place. The philosophy department at the University of South Carolina nurtured and guided me, warned me about the perils of going into professional philosophy, and then helped me prepare for it. A few people deserve special mention: Barry Loewer, who now teaches at Rutgers, was my mentor and is still my friend. Bob Mulvaney remains one of my role models and is the best teacher I have ever had. Davis Baird introduced me to works in philosophy that later turned into part of my dissertation; years later he invited me to give a talk on my work vwhich helped focus my ideas. Ferdy Schoeman combined a dedication to the philosophical life with social activism and was a role model for everyone who knew him. I saw Ferdy two weeks before he died of leukemia in June of 1992; his last words to me were, "We are so proud of you, Catherine-we think you have a very promising future philosophically and otherwise".

Crucial though it may be, intellectual inquiry does not always pay the bills. I wish to thank Gary Dryfoos of IS/CSS (ne6 Project Athena) for hiring me as a minicourse instructor despite the fact that, at the time, my relationship with computers was fraught with fear and ignorance. Amazingly, Carla Fermann also gave me a job, this time as a consultant. They, along with Jeanne Cavanaugh, Tawney Wray and others, have been helpful, supportive, understanding and most indulgent of me.

Formatting this dissertation would have been an onerous task had it not been for the expertise of Jeff Tang; Thanks, Jeff, for helping make my bibliography the

fanciest one on the block.

I also worked at Bentley College, and the faculty and staff there have been per-sonally and philosophically helpful. Special thanks go to Michael Hoffman, Bob Frederick, and Sally Lydon.

While pursuing the degree that would mark the beginning of my life's work, I found (quite by accident) another activity that has become a source of creative satisfaction and income-yes, I am talking about tap dancing. I want to thank the folks at the Leon Collins Tap Dance Studio for being a second family to me; they accepted me regardless of how my thesis work was going. They tolerated (or ignored) my constant complaining, and taught me to express myself artistically. In particular I want to thank Josh Hilberman, Pam Raff, Julia Boynton, Sue Ronson, and Dianne Walker, who have taught me much about music and myself. I made many friends there who have been fun and supportive, among them Linda Pompura, Pat Merritt, Rose Giovanetti, Eve Agush, Josh Hlberman, and Charlie Borden.

My friends have been my surrogate family for a long time. I could not have

completed this dissertation without them. I want to thank Norah Mulvaney for being there for me year in and year out, always honest and loving. Marin Farach was my thesis enforcer, a thankless job that only a long-time friend would be willing to undertake. Thanks for your support and encouragement, Martin; I look forward to working and playing together for years to come. Thanks to Mike Wolfson for being generous with his time and his car (I will always have a soft spot for that red Honda). His firm conviction that everything would turn out right kept me going. Deborah Savage has been loyal, reliable, a great roomate (even though I will never figure out just how she managed to blacken that stainless steel bowl of mine), and a wonderful friend- thanks, Deb. Eric Chivian needs his own category, but I will just say thanks to him for seeing me through such a difficult process.

Since all of my friends, both casual and close, have borne the burden of helping Catherine get her thesis done, I will just say a big thanks to them all, and know that I owe a serious karmic debt to the world.

I left my biggest debt of gratitude to the end. My family has never lost faith that I would succeed (even when I did), and has backed up that faith with support, love, constant reassurance, and a substantial amount of currency. Thanks and I love you Mom, Dad, Nanny, Papa, Elizabeth, John, Clare, Billy, Cathy, Evans, Pat, Winifred, Will, Sam, Pierce, and Xina. And that is not even my extended family!

That leaves one task to dispatch: I am dedicating my dissertation to my sister Elizabeth. She is my best friend, my constant ally, a fair critic, and source of fun, silli-ness, and joy. I look forward to sharing successes, failures, joys and disappointments with you for the rest of our lives. I dedicate this to you with love and respect.

Contents

1 Is Mathematical Knowledge A Priori: Responses to Kitcher's

Skep-ticism 11

1.1 Introduction ... ... 11

1.1.1 Epistemological Questions ... 13

1.1.2 A Preliminary Psychologistic Account of Knowledge ... 16

1.1.3 A Psychologistic Account of A Priori Knowledge ... 17

1.1.4 Kitcher's Account of A Priori Warrants ... 21

1.1.5 Kitcher's Challenge and the Apriorist Response ... 22

1.2 Kitcher's Attack on Mathematical Apriorism ... 23

1.2.1 The Role of Proofs in Mathematical Knowledge ... 23

1.2.2 Kitcher's Account of Proof ... . 29

1.2.3 More Challenges to A Priori Knowledge ... 36

1.3 A Case Against Kitcher's Views on Challenges to Knowledge ... .. 41

1.3.1 Introduction ... 41

1.3.2 The Study ... 42

1.3.3 Applying Asch's Study to Kitcher ... 47

1.4 Final Comments ... ... 49

2 Church's Thesis: a Case Study for Lakatos' Philosophy of Mathe-matics

2.1 Introduction ... a .... 2.2 Lakatos' View of Mathematics as Fallible ...

2.2.1 Some Preliminaries: Terminology, Taxonomy ... 2.2.2 Mathematics is Quasi-Empiricist ...

2.2.3 Fallibilism as a Philosophy of Mathematics ...

2.3 Church's Thesis- A Case Study for Fallibilism ... I...

2.3.1 Introduction ...

2.3.2 Church's Thesis...

2.3.3 Philosophical Arguments in Favor of Church's Thesis ... 2.3.4 An Argument Against Church's Thesis .... ...

54 54 57 57 63 72 74 74 76 80 89

2.4 Concluding Remarks: The Plausibility of Fallibilism as a Working

Phi-losophy of Mathematics ... 97

3 Surveyability and the Four Color Theorem 99 3.1 Introduction ... ... ... o 99 3.2 History of the Four Color Theorem... 101

3.2.1 Kempe's Attempted Proof ... 102

3.2.2 20th Century Developments on the Four Color Theorem . . 104

3.3 Computer Facts about the Four Color Theorem ... . ... 106

3.4 How the Four Color Theorem Challenges the Classical Conception of Proof . . . ... .. . .. . . . 106

3.5 Thomas Tymoczko on the Four Color Theorem ... 108

3.6 Objections to Tymoczko's View ... ... 114

3.6.1 Teller's Comments on Surveyability ... 114

3.6.2 Experiment and Mathematical Proof ... 117

3.6.3 More on Surveyability-Has the Proof Really Been Surveyed? 122 3.6.4 Another Classical Defense of A Priori Proof ... 123

3.6.5 A Computer Proof Predating the Four Color Theorem . . 124

3.7 What is the Epistemological Status of Computer Proofs in General? . 128 3.7.1 Probabilistic Methods in Computer Proofs ... ... .. 128

3.7.2 Does the Use of Probabilistic Methods Alter What Counts as a Proof? ... ... 130

List of Figures

3-1 Kempe's unavoidable set of configurations ...

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103

3-2

a sample 6-ring ...

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105

Chapter 1

Is Mathematical Knowledge A

Priori: Responses to Kitcher's

Skepticism

1.1

Introduction

What is a priori knowledge? Immanuel Kant was responsible for providing philoso-phers with an account that has turned out to be both a guiding principle and philo-sophical conundrum for hundreds of years; he wrote "we shall understand by a priori knowledge, not knowledge which is independent of this or that experience, but knowl-edge absolutely independent of all experience".1 Turning these words into a plausible account of a priori knowledge has proved an arduous task.

Nonetheless, we do have some intuitions about what kinds of knowledge should be a priori on any reasonable account of a priori knowledge. On standard accounts, math-ematical knowledge is held to be a priori, necessary, certain. Part of the explanation

for this view is that the processes by which we come to know truths of

mathematics-following proofs-offer special guarantees that other types of processes (in particular,

perceptual ones) do not offer. But what is it about these processes that entitles us

to claim these guarantees? Exactly what are these guarantees?

Philip Kitcher, in his book The Nature of Mathematical Knowledge

_{lays out}

what he sees as two ways of doing epistemology; he calls them apsychologistic and

psychologistic approaches to epistemology. He points out various problems with the

apsychologistic view, and maintains that doing epistemology psychologistically gives

us the best chance for explaining the nature of knowledge in general, and a priori

knowledge in particular. Once he has laid out a conception of a priori knowledge that

fits his constraints on an adequate theory, he proceeds to give us reasons to think

that mathematical knowledge might not be a priori after all. He seems to think that

the guarantees we need for a process to qualify as a warrant for a priori knowledge

are blocked by a number of challenges in cases of mathematical knowledge.

Proponents of a classical view of mathematical proof should take his charges

seri-ously; if we agree with his general approach to epistemology, then we must examine

closely the processes by which we come to follow proofs. It is with that project in

mind that we will come to see that in fact Kitcher's requirements for a priori warrants

are far too stringent; while it is important that the process of say, following a proof,

be immune to certain recalcitrant experiences, we must distinguish between

recalci-trant experiences which offer reasons and experiences which merely undermine my

confidence in the theorem proved. It is to be hoped (by this author) that in fending

off Kitcher's attack on the a priori nature of mathematical knowledge, we will

illumi-nate Kant's famous words, and uncover some assumptions about the a priori so that

it follows that (at least most) mathematical knowledge is a priori.

1.1.1

Epistemological Questions

Kitcher

says

that there have been two approaches used by philosophers to characterize

knowledge. Before the end of the nineteenth century, he says that many philosophers

used what he calls a psychologistic approach to epistemology

. For them, whether

a belief state was a state of knowledge depended on how the belief was produced.

Of course they supposed that knowledge was a state of true belief; what made it

knowledge was that the processes engendering belief, consisting of events both internal

and external to the subject, were of the appropriate kind. They saw their work in

epistemology as specifying what kinds of processes were the right ones for engendering

knowledge. Kitcher seems to be describing a version of reliabilism--the view that

reliably generated true beliefs constitute knowledge, even though the believer may be

ignorant of the process engendering the belief.

According to Kitcher, the twentieth century ushered in a new view about what

constitutes knowledge, a view which denies that psychological processes have any

relevance for whether a state is a state of knowledge. Kitcher calls this approach

ap8ychologistic because its proponents consider "knowledge [to

be]

differentiated from

true belief in ways which are independent of the causal antecedents of a subject's

states"

_{What is important is the logical connections among a subject's beliefs; If a}

subject's belief that p is "connected in the right way" to certain other beliefs, the

subject knows that p".

Kitcher appears to be giving an account of foundationalism in epistemology.

Ac-cording to this view, certain beliefs-foundational ones-are justified because of some

intrinsic quality (e.g. being analytic) of the belief itself, even though the subject may

3_{Kitcher does not mention any philosophers specifically, but this description could apply to Locke,}

Hume, among others. A [Kitcher 1984], p.4 1.

5_{Accounts varied, but in particular, p had to be "connected" to its logical consequences; if, say,} p implies q and I know that p, then I should also know that q.

be ignorant of the existence of this intrinsic quality of the belief.6

Kitcher objects to apsychologistic epistemology because he be'L ves it ignores some fundamental questions about mathematical knowledge, like "how do we know the axioms of mathematics?" He says that apsychologistic epistemologists often attribute to axioms a special status; they are called 'self-evident', 'a priori', 'analytic'. To attach these labels to mathematical propositions does not, in Kitcher's view, answer his question. His opponents would argue that these distinctions do help by separating the epistemological status of the axioms from the ways we come to know them. We shall consider this issue in greater detail later. However, we should note here that both foundationalism and reliabilism share the feature that whether a belief counts as knowledge relies on facts about which the believer may be ignorant.

Kitcher points out that other philosophers (notably Gilbert Harman and Alvin Goldman') share his dissatisfaction with apsychologistic epistemology. They hold the view that knowledge depends crucially on having the right kind of process producing belief. Apsychologistic accounts of knowledge are flawed in that we can cite cases in which a subject may have a true belief, backed up by excellent reasons and the correct logical connections to other beliefs, but the circumstances under which the subject acquired his belief were defective in some significant way, thus precluding knowledge.

So, even if a belief is say,a necessary truth, if it is arrived at by some unreliable

method, then according to the reliabilists, it would not count as knowledge.

Kitcher says that Gettier examples show how the foundational approaches to knowledge are flawed:

Suppose that X comes to believe that p, p is true, but X's reason for believing that p is not the "right" kind of reason. Consider the following case. Jane sees Joan driving a black car, and comes to believe that Joan owns a black car. Joan does own a black car, but she happened to be driving Janet's black car when Jane saw her. So,

6 [Clay and Lehrer 1989], p.xi.

Jane's reason for believing that Joan owns a black car justifies her belief, but it is not sufficient for knowledge.

The literature on this topic is well-known and suggests that justified true belief is not constitutive of knowledge. Kitcher is using Gettier problems to attack internalist theories of knowledge here; internalism attributes knowledge based on the internal features of beliefs (how they are connected to each other) rather than "the relationship

between the belief and what makes it true".8 He uses this set of problems to reject the apsychologistic view and instead focuses on how beliefs are acquired.

Kitcher suggests that their lack of attention to the psychological processes en-gendering belief created problems for the apsychologistic epistemologists, especially when they tried to give a characterization of a priori knowledge. He points out the problems in one account, given by A.J. Ayer9, who suggested a way to define a priori knowledge:

X knows a priori that p iff X believes that p and p is analytically true.

Kitcher says that it follows from the above account that if we can show that mathematics is analytic (which is no small task), then we can say that we know a priori all statements of mathematics we believe. But of course this is not a correct conclusion. I could come to believe a mathematical statement in an unacceptable way- suppose I come to believe the Pythagorean theorem by dreaming about it, or hearing it from an unreliable source. My belief would not count as knowledge, let alone a priori knowledge.

Of course Ayer could respond to Kitcher's charge by saying that a priori is not the basic notion here, rather analyticity is. It could be that there is a class of propositions, all of which I know a priori just by coming to believe them. In this case Ayer is distinguishing between one's reasons for believing something and the evidence for its truth.

s [Clay and Lehrer 1989], p.xi. 9 [Ayer 1946]

Kitcher notes that apsychologistic epistemologists tried to improve their formuulations-to make mnore sophisticated versions of justified true belief-hut he says all of them failed for similar reasons: "Our success [in defeating apsychologistic proposals for knowledge] results from the fact that the mere presence in a subject of a particu-lar belief or a set of beliefs is always compatible with peculiar stories about causal antecedents"'0. Kitcher seems completely convinced that Gettier examples preclude the possibility of any correct apsychologistic characterizations of knowledge. The only adequate approach for him is to use completely psychologistic principles which count as knowledge beliefs produced only by certain kinds of processes.

1.1.2

A Preliminary Psychologistic Account of Knowledge

Kitcher introduces a simple psychologistic account of knowledge. He uses the term 'warrant' to refer to those processes which produce belief "in the right way".- His analysis follows:

X knows that p iff p is true and X believes that p and X's belief that p was caused by a process which is a warrant for it 2.

Filling out the theory requires specifying conditions on warrants. Most important to showing that a process is a warrant is showing that, given that some process caused a belief, it also functioned to warrant that belief. Background conditions- features of the world both external and internal to the subject's psychology- can affect the warranting power of a process. Given background conditions, a process may not qualify as a warrant for some belief.

Kitcher offers an example from perception.'3 Suppose I am looking at some flowers

'o [Kitcher 1984], p.1 6. 11 [Kitcher 19 8 4],p.16.

'2Kitcher adds that 'process' refers to a token process, a specific sequence of events- not a process type. Two processes can both belong to the same type but not both warrant belief that p, given different background condition.

on a table under normal conditions. I come to believe that there are flowers on the table. According to any reasonable theory of knowledge, perception counts as a process that can potentially warrant belief. Now suppose that on some other occasion, the flowers on the table are surrounded by high quality fake flowers which I cannot distinguish from real flowers. It is possible that I underwent the exact same process in both cases (I saw them the same way in similar light, etc.). But, because I cannot tell the real flowers from the fake ones I cannot now be said to have knowledge that there are real flowers on the table, whereas in the former case I could. So even processes which are potential basic warrants do not function independently of other beliefs;

background conditions affect the warranting power of a process.

This example is a standard one in epistemological literature; it is used to point out the contextually relative and sensitive nature of processes; this assumes a strongly externalist view of knowledge, as facts external to the believer of which she may be ignorant may influence the warranting power of the process by which she comes to

hold a belief.

Kitcher says that the same process can be a warrant at some but not all times, depending on background conditions. He will need to fill in the details of how the warranting process works, what background conditions affect the warranting process, and how that process is affected. We will see later that in his account of priori war-rants, he maintains that sufficiently many background conditions interfere with the warranting processes involved in acquiring mathematical knowledge so as to preclude its being a priori.

1.1.3

A Psychologistic Account of A Priori Knowledge

Now that Kitcher has outlined a general psychologistic approach to knowledge, he turns to the special case of a priori knowledge. 'A priori' applies to an item of knowledge. To say that I know a priori that p is to say that a certain kind of

process caused my belief that p. So, to say that mathematics is a priori is to say

how we come to know mathematical statements. But what kinds of processes are

a priori processes? In particular, what kinds of processes are a. priori warrants for mathematical knowledge?

Kant' well-known explication of a priori knowledge, (given at the beginning of this paper) leaves a lot unclear. Especially vague is the phrase "independent of all experience". It is ambiguous- it could mean "independent of all knowledge" or "independent of any particular item of knowledge".

For purposes of making this more clear, Kitcher uses a standard interpretation of Kant: an item of knowledge is a priori if any experience which would enable us to acquire the concepts involved would enable us to have that knowledge. To make this explicit, Kitcher introduces some terminology.

Let X's experience at t be her sensory state at t. X's sequence of experiences she has had up to t is X's life at t. A life is sufficient for X for p iff X could've had that life and gained sufficient understanding to believe that p'4.

Kitcher uses this terminology to give the following definition of a priori knowledge: X knows a priori that p iff X knows that p and, given any life sufficient for X for p, X could've had that life and still have known that p.

Kitcher note that this account will not work; it is far too weak a formulation. A lot hinges on how we interpret the modality "could've". Does it mean that X does not actually have to have a life in which X acquires the appropriate concepts? This account still does not seem to guard sufficiently against defective ways of belief acquisition. Furthermore, it would seem to follow from Kitcher's first formulation that I could know a priori e.g. that violet is darker than blue-that statement could be analytic (on some accounts), and if so, then in any life in which I acquired the relevant concepts, I would have come to believe it. This argument would also work

on universal empirical knowledge- that there are bodies, etc.

Also, a proper formulation of a priori knowledge should distinguish between empir-ical knowledge of propositions that can be known a priori and true a priori knowledge; we have to count as two different processes the cases in which e.g. I come to know 2+2=4 by proving it and by counting small piles of rocks. What we need to do in order to characterize true a priori knowledge is to specify the ways we actually come to know a proposition a priori.

Kitcher gives an improved version:

X knows a priori that p iff X believes that p, p is true, and p was produced by a process which is an a priori warrant for it.

Kitcher has shifted the burden of defining a priori knowledge to the definition of an a priori warrant. Recall that warrants are psychological processes resulting in beliefs. But what kinds of processes count as a priori warrants? Clearly, perception is ruled out, but what does qualify? Kitcher gives no examples of his own but offers Kant's use of pure intuition (with respect to geometry) as a candidate. He does not explain what he thinks our intuition is, but assumes it works roughly in the following way. Using pure intuition, we (roughly) create a mental picture of, say, a triangle, inspect it, and make judgments about its qualities. What is important to isolate is exactly what makes that process an a priori warrant.

Kitcher says there are three conditions on a process which purports to serve as an a priori warrant:

1. it must produce warranted belief independent of experience. 2. it must produce true belief independent of experience.

3. the same type of process must be available independent of experience.

It is unclear what Kitcher wants from 3. A number of things are left vague. Does he mean the same type of process to be available independent of all experience of just

any particular experience? Surely he does not mean the former. As for the latter option, he is obligated to provide us with a coherent explication.

Kitcher does not want to confine a priori knowledge to necessary truths; he would like to maintain the possibility of contingent a priori knowledge. He does not give any examples here, but presumably he not not want the apriority of mathematical knowledge to hinge on its necessity.

If all a priori truths were necessary, then 1. would follow, says Kitcher. No matter what experiences we had, our mathematical beliefs would be warranted. Why is this so, if the warranting power of a process is affected by contextual information or background conditions? The answer goes roughly as follows:

Everything necessary that is known a priori has the following property: every process that is a warrant for it is an a priori warrant. Why? Because from its necessity we know that there are no possible worlds in which it is false, so there is not a counterfactual situation in which something which was a warrant for a belief ceases to be one. The only ways that a warrant could lose its warranting powers would be if 1) the contextual information changes; or 2) there are worlds in which the belief is false. So, if 2+2=4 is a necessary truth, then it seems to follow that any process I went through to arrive at that belief would be an a priori warrant for it if it were a warrant at all.

Kitcher wants very strong requirements on a priori knowledge. He says a priori warrants should be "ultra-reliable- they never lead us astray"." He adds that it should follow from his account that "in a counterfactual situation in which an a priori warrant produces the belief that p, then p".1e

" [Kitcher 1984], p.24.

1.1.4

Kitcher's Account of A Priori Warrants

From these considerations Kitcher gives the following analysis of a. priori knowledge:

2. X knows a priori that p iff X knows that p and X's belief that p was produced

by a process which is an a prioii warrant for it.

3. A is an a priori warrant for X's belief that p iff A is a process such that, given

any life e, sufficient for X for p

(a) some process of the same type could produce in X a belief that p;

(b) if a process of the same type were to produce in X a belief that p, then it

would warrant X in believing that p;

(c) if a process were to produce in X a belief that p, then p.

In the above account, Kitcher often refers to types of processes. To understand what he means here, we need to know how to specify what a process is and how to divide them into types. Kitcher defines a process as the terminal segment of the causal ancestry of a belief, restricted to states and events internal to the believer. Otherwise, he says, the process would not be available independent of experience.

In the interests of neutrality, Kitcher does not give a specific taxonomy for type-identification of processes. He does say, though, that our intuitions provide some guidance for dividing them. It is obvious to us that some ways of acquiring beliefs are different from others. For example, hearing a statement of the Pythagorean theorem from one's grandmother and following a proof of it clearly should count as different ways of coming to believe that the sum of the square of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the two shorter legs. So these two processes should count as belonging to different types.

Naturally, how fine-grained a distinction we make between types will vary, de-pending on the context. However, Kitcher warns that some type-division proposals

would flout any of our principles of taxonomy. Any taxonomy which counts e.g. both the process of following a proof of a theorem and also hearing it from your grand-mother as being of the same type Ehould be disallowed. Even though Kitcher claims the theory is neutral, it must not violate our intuitive principles for what count as dissimilar ways of forming beliefs. The principles he has in mind may turn out to influence how we type-identify processes.

1.1.5

Kitcher's Challenge and the Apriorist Response

Kitcher sees his psychologistic framework as constraining the apriorist program in important ways. If the apriorist philosopher is to succeed in making a priori knowledge a useful notion for epistemology, then she must follow the form he has specified, and then fill in the details. She must specify processes according to the restrictions in 3) and give type-identity conditions which conform to some principles of classification which are he says are standardly used in dividing processes of belief-formation.' 7 If

her account of a priori warrants has satisfied 3), she succeeds; Otherwise her case for the existence of a priori knowledge has failed.

I intend to meet this challenge, not by satisfying Kitcher's requirements, but rather by showing that his analysis places unrealistic constraints on what counts as a priori knowledge. I will show that the conditions on a priori warrants in 3) result in nothing being a priori, which is a problem. While there are plenty of reasons to object to a classical notion of the a priori, Kitcher's approach tries to provide for an account of a priori knowledge; failing to show that anything satisfies his account makes his entire epistemological approach less plausible. Also, I will show that there are reasons to believe that within Kitcher's framework his arguments against a priori knowledge work equally well against all knowledge. So if he is successful, he will have eliminated the possibility of any kind of knowledge of mathematics, not just a priori knowledge

of it.

1.2

Kitcher's Attack on Mathematical Apriorism

Kitcher sets up his psychologistic framework to include the notion of a priori war-rant so he can attack what he calls "mathematical apricrism". According to him, proponents of mathematical apriorism consider mathematical knowledge to be a pri-ori knowledge. Since most statements in mathematics are justified by use of proofs,

Kitcher focuses on what he thinks is the traditional notion of proof. He tries to show

that the process (or processes) of following a proof does not meet the requirements

for a priori warrants.

1.2.1

The Role of Proofs in Mathematical Knowledge

Kitcher begins his examination by looking at how we standardly characterize proofs. He objects to what he calls a structural conception of proofs- the view that a proof

in a system is a sequence of sentences in the language of the system sach that each

member of the sequence is either an axiom of the system or a sentence which results from previous members of the sequence in accordance with some rule of the system. He thinks that it is presumptuous to think that proofs in standard formal systems

are the only acceptable kind of proofs.

Other criteria enter into our decisions as well, like acceptance by the mathematical

community. Kitcher's point is well-taken but there are reasons to think that the mathematical community accepts proofs at least in part because they are of a standard form. Without standards of formal rigor, it would be much more difficult to tell

whether a proof was acceptable. Also, it is possible that without such standards mathematicians would have more dispates over whether something was a proof.

we have become more rigorous and advanced. Our proofs are not written in the language of first-order logic; they are abbreviations of formal proofs. What counts as a formal, rigorous proof or an informal, abbreviated proof is dependent on the community. Given the fact that the community changes constantly, what makes out current proofs "genuine" proofs?

Given that acceptable proofs are informal abbreviations, dependent on the audi-ence, and in a constant state of change, what makes them "genuine" proofs?

Kitcher says that apsychologistic epistemologists answer Kitcher's question in the following way: genuine proofs are those whose axioms are "basic a priori principles" and whose rules of inference are just those that are "elementary a priori rules of inference".18 But to say this is not to give a complete explanation, Kitcher responds, unless accompanying it is a thesis about how these principles can be known, and how we can use these rules of inference to extend our knowledge[p.37], a thesis which must be detailed and well-argued.

A better way, says Kitcher, of characterizing proofs within the framework of an

adequate epistemology is to give a functional definition. Proofs are sequences of sentences that serve a certain purpose for us. But what purpose? Kitcher says the apriorist would have to explain the purpose as follows:'9

proofs codify psychological processes which can produce a priori knowl-edge of the theorem proved. Similarly, to follow a proof is to engage in a particular kind of psychological process which results in the acquisition of a priori knowledge.

What does it mean to say that a psychological process can produce a priori knowl-edge? A number of psychological processes can result in e. g. believing that 2+2=4, but only certain ones count as following a proof. Kitcher's characterization must

18 [Kitcher 1984], p.3 7. ' [Kitcher 1984], p.3 7.

specify what kinds of processes are the right ones in order to separate a priori from a posteriori knowledge.

To clarify what he means here Kitcher introduces some terminology. A statement is a basic priori statement "if it can be known a priori by following a process which is a basic warrant for belief in it".20 _{Recall that basic warrants are processes which}

involve no other beliefs, according to Kitcher. So far he has offered no examples of processes that might qualify.

Kitcher says that proofs must begin from basic a priori staten ents. Further state-ments result from applications of apriority-preserving rules of inference. A rule is apriority-preserving just in case "there is a type of psychological process, consisting in transition from beliefs in instances of the premise forms to the corresponding in-stances of the conclusion form, unmediated by other beliefs, such that, if the inin-stances of the premise forms are known a priori, then the transition generates a priori knowl-edge of the instance of the conclusion form.""21 We are still left not knowing exactly what he has in mind. For example, it is unclear how his analysis would explain how we use e. g. modus ponens.

Using these psychologistically defined terms, we can now define proof as Kitcher thinks the apriorist should:2 2

To follow a proof is to undergo a process in which, sequentially, one comes to know the statements which occur in the proof, by undergoing basic a priori warrants in the case of basic a priori statements and, in the case of those statements which are inferred, by undergoing a transition of the type which corresponds to the rule of inference in question.

Characterizing proofs functionally as well as structurally gives us insight into

what purposes proofs serve in mathematics. While it is true that what we mean by

2o [Kitcher 1984], p.3 8.

21 [Kitcher 1984], p.38.

'proof' is 'proof in a standard formal system with a certain form...', that does not completely explain why we consider those particular sequences to be proofs. What makes them proofs is that they do a certain job- they convince us of the truth of the theorem proved, using clear, explicit, accepted reasoning. Proofs serve a prescriptive, normative function. If I have followed a proof of the Pythagorean theorem, then I can conclude with impunity that whenever I do computations involving right triangles, if I add the squares of the lengths of the two shorter legs, the sum will equal the square of the hypotenuse. Following a proof of a theorem gives me good reasons to believe that it is true, and these reasons justify my belief in the theorem. In fact, following

a proof compels my belief in the theorem.

The mathematical apriorist would readily agree that proofs are distinguished by the fact that they increase our mathematical knowledge. Since proofs begin with basic a priori principles and proceed using apriority-preserving rules of inference, one can follow a proof and extend his knowledge without adverting to experience; that

is, once he understands the concepts involved, his resulting knowledge is warranted

or justified, matter what kind of experiences he has had.

For Kitcher, "Psychological processes" refers to internal causal processes of the subject. They must be internal since their production warrants a priori knowledge. For a proof to "codify" or pattern psychological processes there should be kind of

correspondence between the steps in the proof and the steps in the processes.

On standard accounts, The activity of following a proof of a theorem involves engaging in a process that results in acquisition of a priori knowledge. Kitcher is right to point out that we are owed an account of what that process is. The apriorist could respond that the notion "following a proof" is a primitive notion, but it is in fact a complicated process. Consider the following example: I come to believe p based on my belief of p and q. If we examine the individual steps, we see that there is a transition in my belief state that is somehow brought about by the previous step.

Whatever makes me make the conclusion is the process; it is the relation between the step and the transition.

For Kitcher's analysis to be successful, he must also explain how proofs codify psychological processes. Roughly speaking, we do engage in certain mental activities when we follow the steps in a proof, but there is no reason to believe that there is

La 1-1 correspondence between the steps in a proof and the psychological processes

we undergo. Kitcher offers no suggestions about how to translate steps in a proof into psychological processes. And we do not have any intuitions about how many discrete psychological processes we undergo in applying the rule of modus ponens, for example. Whereas Kitcher rightly points out that we do undergo some psychological transitions when we follow proofs, his analysis leaves the details of how this works unexplained. Of course, so does the apriorist, but that means that his account does not provide more explanatory power than the standard view.

If the psychologistic epistemologists are right, then the best way to answer the question "what job do proofs do?" is to be had by looking at what we do when we follow proofs, and how proofs reflect mental processes we undergo.

This investigation could prove helpful for answering questions in epistemology. For example, if psychologists discovered that the processes corresponding to steps in a proof all required perceptual mechanisms, then a case could be made that mathe-matical knowledge is empirical. On the other hand, if experiments determined that mental processes in the practice of doing mathematics were just the instantiations of logical principles, starting from logical axioms, then that would be evidence that mathematical knowledge is knowledge of logic. Or if psychological data show only non-empirical mechanisms at work in calculation, that we would be more inclined to consider mathematical knowledge a priori.

Kitcher's analysis of proof interpreted charitably might, with the appropriate ac-companying data, yield some hypotheses about how our mathematical practices work.

But, it fails to take into account the job proofs do- they are arguments, giving us sufficiently good reasons to believe that a theorem is true. Kitcher never links the notion of proof to the justification of the truth of a statement; or, more important for his project, to justification of one's knowledge of mathematics.

Kitcher clearly states that he is rejecting the apsychologistic rendering of knowl-edge as justified true belief for the reason that the account fails to disqualify as knowledge cases of true belief acquired in some epistemically defective way. Gettier-type cases give ample intuitive evidence for the need for an appropriate causal story if a true belief is to count as a state of knowledge. But in the case of mathematical knowledge, there are two stories to be told: first, some account of how we acquire knowledge of mathematics; second, an explanation of how proofs serve to justify our beliefs that many mathematical statements are true, how proofs reveal the deduc-tive structure of mathematics, and how we can use them to extend our knowledge of mathematics.

Frege was interested in working out the details of the second story. He thought epistemology concerning mathematics should definitely be apsychologistic.23 Frege was disturbed that some mathematicians "confuse the grounds of proof with the mental or physical conditions to be satisfied if the proof is to be given".2 4 He cites one of his favorite examples from the literature of his time Schroeder's "Axiom of Symbolic Stability. It guarantees us that throughout all our arguments and deductions the symbols remain constant in our memory- or preferably on paper".2 5 _{That psychology}

could affect the foundations of mathematics to the extent that we needed safeguards against mysteriously changing variable letters seemed absurd to Frege. What he thought affected the foundations of mathematics was the degree of rigor with which many results were formulated.

"8In "Frege's Epistemology", Kitcher defends the view that his psychologism is not the kind to which Frege would have objected.

24_{Grundlagen, p.VIII} "Grundlagen, pp.VIII-IX.

Frege acknowledges that much of mathematics seems self-evident. To reqiire a proof of 2 + 2 = 4 is "almost ridiculous". But proofs for Frege do more than just establish the truth of a theorem: "the aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another".2B Proofs hold the key to mathematical advancement. By doing them we learn the limits of application of techniques and concepts. Proofs uncover part of the deductive structure of mathematics. Knowing where a theorem fits within this structure helps us decide where and when to look for new theorems.27

Kitcher's account also does not provide an explanation of how proofs lead us to

mathematical discoveries. The apriorist has no reason to accept his notion of proof,

for it fails to capture key aspects of proof- its use in justification, its use in extending our knowledge. She can concede the benefits of a causal account in explanation of the origins of mathematical knowledge. But Kitcher's story may turn out to be insufficient for purposes of doing work in foundations of mathematics.

1.2.2

Kitcher's Account of Proof

Now Kitcher is ready to provide the following thesis about the form of apriorist proof:2 8

4)there is a class of statements A and a class of rules of inference R such that: a) each member of A is a basic a priori statement; b) each member of R is an apriority-preserving rule;

c) each statement of standard mathematics occurs as the last member of a

se-quence, all of whose members either belong to A or come from previous members in

eGrundlagen, p.2.

7By "structure", I do not mean logical structure. I merely use the term to refer to whatever organisation exists in various fields of mathematics. No logicistic assumptions are intended.

accordance with some rule in R.

He says that whereas 4c) has been traditionally regarded as controversial, 4a) and 4b) have been accepted without question. Since the goal of his program is to uncover apriorist assumptions about mathematical knowledge, Kitcher considers 4a) and 4b) just as suspect as 4c). Kitcher makes two criticisms of mathematical apriorism. The first attacks claims which are instances of 4a); the second examines the apparent incompatibility between the supposition that many theorems of mathematics can be known a priori and the fact that some of these theorems can be proved only by demonstrations of great length.

Kitcher attacks 4a) first, saying that apriorists have committed themselves to the existence of basic a priori statements, which he thinks is a terrible mistake; processes traditionally regarded as a priori warrants are in fact not, so they cannot call instances of 4a) a priori.2 9 I will first consider a worry directed at the general form of his argument.

Here is the general structure of his argument against 4a):

1')Traditionally, we have regarded many statements as being basic a priori.

2')A necessary condition for basic apriority is being caused by a process which is

an a priori warrant for it.

3')Many statements which have been traditionally regarded as basic a priori are not caused by processes which serve as a priori warrant for them.

4')Therefore, such statements are not basic a priori.

Kitcher's argument initially looks reasonable, but it depends crucially on the plau-sibility of 2'). Unless Kitcher's case for 2') is quite persuasive, there is as much reason to conclude not 2') as there is to conclude 4') Why? Well, we have strong reasons to think that that 1') is true- intuitions and philosophical traditions support the fact that many statements are a priori. If Kitcher wants to hold 2'), then he must give 29 [Kitcher 1984], p.39. Kitcher devotes Chapters 3 and 4 to the task of discrediting several theories of knowledge which are committed to the existence of basic a priori statements.

evidence for it that either appeals to our views about a priori knowledge or shows us

how our intuitions were mistaken. Otherwise, the more sensible solution is to

con-clude that a priori warrants are not necessary for a priori knowledge. Later in this

section I will argue that this is just what we should conclude.

Kitcher's second criticism questions the correctness of 4) as a characterization of

a priori proof. Consider his inductive argument:

Let S be any true mathematical

statement. By 4c) there is a sequence of sentences, [which is the proof of S], all of

whose members belong to A or come from previous members by one of the rules in R..

We can show by induction, using 4a) and 4b) that every statement in the sequence is

knowable a priori. A fortiori, S is knowable a priori. Hence every truth of standard

mathematics is knowable a priori.

It follows from the analysis above that if S is a proof, (consisting of basic a priori

principles or following from one by use of an apriority-preserving rule of inference)

then we should be able to come to know S by following a proof of it. That process

(following the proof) will then serve as an a priori warrant for the belief that S.

Kitcher says that the existence of very long proofs of mathematical statements

may threaten the inductive argument. There are theorems whose proofs are so long

that one person could take years to go through one of them. Since I am fallible,

it is possible that such a proof contained some errors that I overlooked. I may not

be completely certain that I followed the proof correctly, and my knowledge of the

statement is therefore not a priori.

Kitcher gives three possible resolutions of this conflict:"'

i) we can accept the inductive argument and the point about long proofs,

con-cluding that no version of 4) can be correct.

ii) we can accept the inductive argument and reject the point about long proofs,

thereby concluding that 4) is sufficient to establish apriorism.

0so [Kitcher 1984], p.4 0.

iii) we can reject the inductive argument, concluding that 4) does not suffice to

establish apriorism.

We know that, for familiar reasons, inductive arguments involving vague predicates are not always valid. Kitcher points out that the worry about long proofs could be because the term 'a priori' is vague. It is possible that the statements encountered early in a proof have a high degree of certainty, but inferences resulting in later conclusions do not preserve certainty. After some point it would not be correct to ascribe knowledge of the conclusion, just as at some point in the process of adding 1 to a number it would not be correct to call that number 'small'. iii) is certainly a

possible explanation of the problem long proofs present, but saying that 'a priori' is

vague does not tell us whether we should reject apriorism; it is unclear what conclusion we should draw if it turns out that apriority is a vague notion.

Apriorists, Kitcher says, will oppose iii) and defend a priori knowledge of the

conclusions of long proofs, adopting ii). Kitcher acknowledges Hume's observation that as we review proofs and others agree that they are correct, we become more convinced of their truth. But the fact that our certainty increases with agreement of our peers is not relevant to the truth of the theorems. Kitcher considers uncertainty about a proof to be incompatible with a priori knowledge of it.

The apriorist will want to separate the psychological feelings of certainty about proofs from the epistemological status of the theorems proved. He suggests two ways to defend ii). We could say that uncertainty stems from the fact that most proofs are informally structured, and formalization would remove any doubt. Another possibility is to propose that we can know a proposition without knowing it for certain.

The first suggestion clearly will not work. Quite the contrary- presenting a proof in formal notation will increase its length enormously, exacerbating the problem. Kitcher correctly notes that some theorems never receive rigorous proofs, even by informal standards. Also, the activity of formalizing proofs is just as subject to

errors, so formalization leaves us in a worse state.

What about the second option? Does rational uncertainty preclude a priori knowl-edge? Kripke thinks not:3 2

Something can be known, or at least rationally believed, a priori, without being quite certain. You've read a proof in a math book; and, though you think it's correct, maybe you've made a mistake. You do often make mistakes of this kind. You've made a computation, perhaps with an error.

Kripke thinks it is a mistake to conflate apriority and certainty. Kitcher does acknowledge a distinction: "One can go easily astray here, by conflating a priori knowledge with knowledge obtained by following a non-empirical process". 33 But

Kitcher disagrees with the view that rational uncertainty is compatible with a priori knowledge. A priori knowledge for him is, in Mark Steiner's words,

"...incorrigible-we could never be justified in giving it up once it is warranted. And perhaps it is even unrevisable- meaning that nothing at all could shake our conviction"34

But Kant never had such stringent requirements for a priori knowledge- for him it was nonempirical and necessary.35 So it looks like there are at least two notions of apriority, but Kitcher chooses to attack the more stringent one38. Mathematical knowledge probably does not meet the requirements for Kitcher's notion of apriority, but that is not surprising. Nor is it disturbing; no one would expect that it should. What is important for the apriorist is that mathematical knowledge be a priori in

Kant's sense, an issue we will examine later.

Kitcher thinks that uncertainty interferes with the a priori warranting process be-cause experiences could yield situations in which e. g. the book in which the theorem

"'Kripke, 1972,p.39 88 [Kitcher 1984], p.4 3

TM_{Steiner, p.452.}

8"Steiner, p.452. 8TSteiner, p.452.

was proved is discredited, or the mathematical community decides to reject the proof.

Kitcher says that when I have doubts arising from following a complicated proof, then

if there are also circumstances under which experiences suggested the falsity of the

theorem (e. g. the book I read was discredited by mathematicians), then I cannot

conclude that I know the theorem a priori. However, this is not the case with

regu-lar warranting processes "because of the kindly nature of background experience".

So, rational uncertainty does not preclude knowledge, but it does preclude a priori

knowledge, which leads Kitcher to conclude i).

Why does Kitcher believe that non-apriori warrants are not also undermined by

rational uncertainty? it seems as if he is saying that even if I am warranted in believing

that p (either a priori or a posteriori), I do not know that p unless background

conditions are "right". Non-a priori beliefs must then be less affected by background

conditions, whereas a priori beliefs are more likely to be affected. This is exactly the

opposite of my intuitions about knowledge.

Kitcher does not argue for this point other than to say that if the quality if our lives

were different, that rational uncertainty would preclude knowledge; since many of our

ordinary beliefs are not undermined by experience (unlike some of our mathematical

beliefs), ordinary knowledge is not blocked. I find this argument puzzling; we have

lots of experiences that undermine our perceptual judgments, but few that undermine

e. g. the belief that 2+2=4. Optical illusions, perceptual infirmities, and poor lighting

are all common examples of how experience can lead us astray. But, we do count very

many of our perceptual beliefs as knowledge. If Kitcher allows rational uncertainty

to interfere with a priori knowledge, which we intuitively count as equally or even

more certain than a posteriori knowledge, then I do not see how he can stop rational

uncertainty from precluding knowledge. Certainly any obvious attempts to solve this

problem would strike me as ad hoc solutions, unless they gave compelling reasons to

explain away and overcome our intuitions.

Kitcher considers the problem with uncertainty undermining knowledge of long

proofs to be the same one Descartes encountered with deductions in the Regulae.

Since extended deductions "exceeded the scope of what we can simultaneously present

to ourselves"", they are uncertain. Descartes' solution involved practicing following

the deduction to ourselves so often that eventually we can apprehend the entire proof

in one mental act. Although Kitcher, like others, views Descartes' solution as

in-feasible because of our physical limitations, he agrees with Descartes' picture of the

psychological process of following a proof.

Kitcher gives a modern version of Descartes' view.

" A proof begins with an

axiom, which I intuit, and from it I infer (using, perhaps, one-premise rules) the next

statement, and from that the next one, and so on. Suppose also that I can present

to mind only one axiom and three inferential steps in the proof. Then I store the

results, recalling them in order to go on following the proof. In Kitcher's terminology,

I undergo a process which is a basic warrant for belief in the axiom. The problem

arises when I no longer believe the axiom on the basis of the original warrant, but

only because I remember having undergone a warranting process. But that

process-the recollection- is not itself an a priori warrant for my belief in process-the axiom, so my

belief is uncertain. So we must conclude that no version of 4) can be correct.

It is true that we cannot represent long proofs to ourselves; few mathematicians

would ever consider that a requirement for knowledge of a theorem. In fact, some

psychological studies have shown that we cannot in general represent more than 7

symbols in short-term memory.

_{If this is the case, then Kitcher's skeptical worries}

about long proofs are ill-founded. Furthermore, his skepticism leaves open the

for

during

ss [Kitcher 1984], p.43.

S[(Kitcher 1984], pp.44-45. "ofind reference for this.